In general, in states which are considered “employment at will states,” employers have the right to fire employees for any reason, no matter how unfair or irrational. However, there may be specific laws or circumstances that create an exception. One example of an exception is if you’re fired because of your race, sex, religion, etc. The only way to know for sure is to contact a lawyer familiar with the laws in your state.
You may be correct that the employer is trying to establish a paper trail to support a termination. The good news is that you are being given clear standards to meet, so you'll know exactly what the employer wants you to do. If you meet those standards, presumably your job will be safe. If you meet the standards and are still terminated, then you'd need to look more closely at the situation to see if any illegal reason was the basis of your termination.
Even if you fail to meet the standards and are terminated, but you have information that other people with your same job were not required to meet the same standard, then you could look to se if there was an illegal motive in targeting you but not others.
As far as the signature on the document, anytime a signature is requested that merely acknowledges that you have recieved and understand the paper, you can be required to sign it or face termination. If the signature indicates anything else, you may want to run it by an attorney before signing. It is perfectly okay to tell your employer that you want a day to review any document you are being asked to sign.